Abelian Group

An abelian group is a group in which the law of composition is commutative, i.e. the group law \(\circ\) satisfies \[g \circ h = h \circ g\] for any \(g,h\) in the group.

Abelian groups are generally simpler to analyze than nonabelian groups are, as many objects of interest for a given group simplify to special cases when the group is abelian. For example, the conjugacy classes of an abelian group consist of singleton sets (sets containing one element), and every subgroup of an abelian group is normal.

An abelian group is a set \(G\) combined with a binary operation \(\circ: G \times G \rightarrow G\) (i.e. \(\circ\) takes two elements of \(G\) and returns an element of \(G\)) satisfying the following properties:

Group Axioms

1) Associativity. For any \(x, y, z \in G \), the relation \( (x \circ y) \circ z = x \circ (y \circ z) \) holds.

2) Identity. There exists an \( e \in G \), such that \( e \circ x = x \circ e = x \) for any \(x \in G \). \(e\) is called an identity element of \(G\). It can also be shown that \(e\) is necessarily unique, and is thus usually referred to as the identity element of \(G\).

3) Inverse. For any \(x \in G\), there exists a \(y \in G\) such that \(x \circ y = e = y \circ x \). \(y\) is called an inverse of \(x\). It can also be shown that \(y\) is necessarily unique, and is thus usually referred to as the inverse of \(x\). It is also customary to refer to the inverse of \(x\) by \(x^{-1}\).

4) Closure. For any \(x, y \in G \), \(x \circ y \) is also in \(G\). This follows from the definition of \(\circ\) immediately, but is worth noting as the construction of subgroups must take this axiom into account.

5) Commutativity. For any \(x,y \in G\), the relation \(x \circ y = y \circ x\) holds.

The first four conditions are the only ones necessary to define a group; the final one (commutativity) is the distinction between an abelian group and a nonabelian one.

The simplest examples of abelian groups are cyclic groups, which are groups generated by a single element and thus isomorphic to \(\mathbb{Z}_n\); recall that \(\mathbb{Z}_n\) is defined as

\( \mathbb{Z}_n\), the set of integers \( \{0, 1, \ldots, n-1\} \), with group operation of addition modulo \(n\).

They are so named because successive application of the group law to the generator forms a cycle amongst the group's elements; e.g. the powers of the generator \(g\) of \(\mathbb{Z}_5\) are \(g^0, g^1, g^2, g^3, g^4, g^5 = g^0, g^1, g^2, g^3, g^4, \ldots\), making the elements \(\{g^0, g^1, g^2, g^3, g^4\}\). Since \(g^ag^b=g^bg^a=g^{a+b}\), these groups are abelian.

Though all cyclic groups are abelian, not all abelian groups are cyclic. For instance, the Klein four group \(\mathbb{Z}_2 \times \mathbb{Z}_2\) is abelian but not cyclic.

In contrast, the group of invertible matrices with a group law of matrix multiplication do not form an abelian group (it is nonabelian), because it is not generally true that \(MN = NM\) for matrices \(M,N\). The symmetric group \(S_n\) is also nonabelian for \(n \geq 3\).

Rings are also examples of abelian groups, with respect to their additive operations. Further, the units of a ring form an abelian group with respect to its multiplicative operation. For example, the real numbers form an additive abelian group, and the nonzero real numbers (denoted \(\mathbb{R}^{*}\)) form a multiplicative abelian group.

As previously mentioned, abelian groups form special cases of many group properties, such as (from the introduction) the fact that the conjugacy classes are singleton sets. Similarly,

• The center of a group (the set of elements that commute with all group elements) is equal to itself. The converse is also true: if the center of a group is equal to the group itself, the group is abelian.
• The commutator (defined as \(g^{-1}h^{-1}gh\)) of any two elements of an abelian group is the identity.
• The derived subgroup of an abelian group is trivial.

Abelian groups also form a variety of algebras, meaning that

• Any subgroup of an abelian group is also abelian.
• Any quotient group of an abelian group is also abelian.
• The direct product of two abelian groups is also abelian.

Finally, the set of homomorphisms from an abelian group to another forms another abelian group, with the group law

\[(f+g)(x) = f(x) + g(x) \quad \forall f,g: G \rightarrow H.\]

Abelian groups can be classified by their order (the number of elements in the group) as the direct sum of cyclic groups. More specifically,

Kronecker's decomposition theorem. An abelian group of order \(n\) can be written in the form \[\mathbb{Z}_{k_1} \oplus \mathbb{Z}_{k_2} \oplus \ldots \oplus \mathbb{Z}_{k_m},\] where the \(k_i\) are powers of primes, and the \(k_i\) multiply to \(n\). This representation is unique up to permutations of the summands.

For example, an abelian group of order 15 can be written only as \(\mathbb{Z}_3 \oplus \mathbb{Z}_5\), implying that all abelian groups of order 15 are isomorphic. An explicit example is \(\{0, 5, 10\} \oplus \{0, 3, 6, 9, 12\}\). There are two additional special cases worth noting:

• A group with order \(p\) is necessarily abelian and isomorphic to \(\mathbb{Z}_p\), and is thus cylic.
• A group with order \(p^2\) is necessarily abelian, and isomorphic to either \(\mathbb{Z}_{p^2}\) or \(\mathbb{Z}_p \times \mathbb{Z}_p\).

Additionally, it follows from Kronecker's decomposition theorem above that the number of non-isomorphic abelian groups of order \(n=\prod_i p_i^{e_i}\) is

\[a(n) = \prod_i P(e_i),\]

where \(P(n)\) is the number of partitions of \(n\); in other words, \(a(n)\) is the product of the number of partitions of each exponent in the prime factorization of \(n\).

• Group theory
• Subgroup
• Ring theory